Global Existence Of Positive Solutions To A Degenerate Reaction-diffusion System

In this paper, we give a brief review of the results to a degenerate reaction diffusion systems and a nuclear reactor model. We consider a degenerate reaction-diffusion system, and prove the global existence of positive solutions to the system. The reaction diffusion system is derived from diffusion phenomenon which widespread in the nature, and it involves the physical, chemical, biological and biological group dynamics, ect. The mathematical model we study in this paper is the following. Here Ω(?)RN(N≥3), is a bounded domain with smooth boundary (?)Ω, m>1,n>1, a>0,b>0,0<T<∞, uo(x) and vo(x) are nonnegative functions. When m, n>1, the system can be used to describe processes of combustion and the spread of fluid flow in porous media. Particularly, when m=n=1, the model describes the processes of nuclear reactor, and the functions u, v can be interpreted as temperature and the speed of neutron flow. At first, in this paper we make a brief review of the relevant background and research status, discussed the application of the model. Then, we discuss the principle, method and basic knowledge. At last we study the model further and obtain the following result.Theorem:Suppose that(1) u0>0, v0>0, x∈Ω,(2) u0(x)=v0(x)=0,(?)u0/(?)η,(?)v0/(?)η<0, x∈(?)Ω and the constants m, n satisfy (m-1)n>0. Then, all the nonnegative solutions of problem (0.1) exist globally.In the research process of this model, first we prove that the positive solutions of prob-lem (0.1) exists locally by means of parabolic regularization in Section4. Due to the asym-metric of this model, the classical comparison principle may be failure, we must verify the comparison principle in the condition of asymmetric. In Section5, we construct a smooth approximation sequence aiε. And take advantage of the following backward equation to prove that the classical comparison principle holds, where Xi∈C0∞(Ω).In the sixth Section of this paper, we construct a upper solution and give the upper bound estimation.Firstly, we define functions Φ(x) and Ψ(x) as the following-△Φ(x)=1,x∈Ω,Φ(x)=0, x∈(?)Ω;-△Ψ(x)=1,x∈Ω,Ψ(x)=0, x∈(?)Ω. and Φ(x),Ψ(x) satisfy (1) When x∈Ω,then Φ(x)>0,Ψ(x)>0,when x∈(?)Ω then (?)Ψ(x)/(?)η<0,(?)Ψ(x)/(?)η<0,(2) M=max{Φ(x)x∈Ω,Ψ(x)x∈Ω<∞. Then,we prove that for certain k,l1,l2,the upper solution to(0.1)can be chosen:u(x,t)=(k(Φ(x)+1))l1,v(x,t)=(k(Ψ(x)+1))l2. At the end,we get our results of global existence by means of Gronwall’s inequality.